Spectra and energy levelsQuestion 150S: Short Answer

Photons and jumps between levels

When an electron in an atom goes from energy E2 to a lower energy E1 a photon of energy E = h f = E2 – E1 is emitted. These questions give practice in using these ideas.

·Planck constant h = 6.63 x 10–34 J Hz–1

·speed of light = 3.00 x 108 m s–1

·magnitude of charge on electron = 1.60 x 10–19 C.

Mercury lamps

Mercury lamps are used in insect killers and in airport security to check passports, because they emit ultraviolet light of wavelength 253 nm, which can ionise air (insect killing) or make certain ‘invisible inks’ glow (security checks). It’s all down to where the energy levels of the mercury atom happen to lie.

Here are some of the energy levels of the mercury atom.

  1. The energies of the levels are given starting with zero for the lowest level. Relative to an electron outside the atom, the energy of the lowest level is – 16.7 x 10–19 J or – 10.4 eV. What is the significance of the fact that these values are negative?
  1. Confirm that the energy of the level at 7.86 x 10–19 J is equal to 4.91 eV.
  1. The 253 nm ultraviolet line comes from a transition between two of the levels shown. Between which pair? In which direction is the jump?
  1. A photon is also emitted when an electron goes from the level at 8.84 eV, to the one at 6.70 eV. What is the wavelength of this photon?
  1. What is the energy of the longest wavelength photon which might be absorbed by an electron jumping between the levels shown? Which way does the jump go?

Sodium street lamps

  1. The bright yellow colour of sodium street lamps is light of wavelength 591 nm. These photons come from electrons going to the ground (lowest energy) state of sodium at – 5.14 eV, from a level at a higher energy. What is the energy of this higher level? Why is the answer negative?
  1. In fact, the higher level is split into two levels at very slightly different energies. What effect will this have on the spectral line?

Hydrogen levels

The energy levels of hydrogen follow a simple scheme. There is a sequence depending on integer numbers n, starting at n = 1. The energy for n = 1 is E1 = – 13.6 eV. The energy for any other value of n is En = E1 / n2. Here is a picture of these levels.

  1. Plot a graph to test whether the values given in the diagram do in fact fit the equation En = E1 / n2. What is wrong with simply plotting En against n? Consider carefully what would be a good graph to plot, so that points are not crowded at one end, and that you expect a straight line. Using a spreadsheet gives you the chance to try different graphs.
  1. The Lyman series of lines are the result of jumps down to the ground (lowest energy) state. Which transition gives the longest wavelength Lyman series line? Show that all the spectral lines in the Lyman series fall in the ultraviolet. The invisible ultraviolet part of the spectrum begins at a wavelength of about 400 nm.
  1. In the light from some quasars, the Lyman line at 120 nm is red-shifted into the visible region. What is the minimum red-shift needed to do this?

The Balmer series of lines falls in the visible region. They can be seen in the spectrum below.

  1. One of the lines falls at 486 nm. What are the values of n for the two levels between which an electron is going?